We prove two theorems related to the Central Limit Theorem (CLT) for Martin-L\"of Random (MLR) sequences. Martin-L\"of randomness attempts to capture what it means for a sequence of bits to be "truly random". By contrast, CLTs do not make assertions about the behavior of a single random sequence, but only on the distributional behavior of a sequence of random variables. Semantically, we usually interpret CLTs as assertions about the collective behavior of infinitely many sequences. Yet, our intuition is that if a sequence of bits is "truly random", then it should provide a "source of randomness" for which CLT-type results should hold. We tackle this difficulty by using a sampling scheme that generates an infinite number of samples from a single binary sequence. We show that when we apply this scheme to a Martin-L\"of random sequence, the empirical moments and cumulative density functions (CDF) of these samples tend to their corresponding counterparts for the normal distribution. We also prove the well known almost sure central limit theorem (ASCLT), which provides an alternative, albeit less intuitive, answer to this question. Both results are also generalized for Schnorr random sequences.

翻译：我们证明与随机(MLR)序列的 Martin-L\” 中限参数(CLT) 有关。 马丁- L\“ 随机性” 试图捕捉一个位子序列“ 纯随机” 的含义。 相比之下, CLT 并不对单个随机序列的行为做出断言, 而只对随机变量序列的分布行为做出断言。 生动地说, 我们通常将CLT 解释为对无限多序列的集体行为的表示。 然而, 我们直觉是, 如果一个位子序列是“ 绝对随机的 ”, 那么它应该提供一个“ 随机性源 ”, 而对于 CLT 类型的结果应该持有。 我们通过使用一个从单一的二进制序列中产生无限数量的样本的抽样方案来解决这一困难。 我们显示, 当我们将这个方案应用到一个随机序列的 Martin- L\\ 的 分配过程时, 这些样本的经验时刻和累积密度函数 倾向于对应的正常分布的对应方 。 我们还证明一个众所周知的非常肯定的核心限制 点( ASCLT) 的答案是普通的答案。